Characterizations of Moore-Penrose inverses of closed linear relations in Hilbert spaces

Abstract: This paper examines the Moore-Penrose inverses of closed linear relations in Hilbert spaces and establishes the result $\rho(\mathcal{A}) = {\lambda \in \mathbb{C}: \lambda^{2} \in \rho(TT^{\dagger}) \cap \rho(T^{\dagger}T)}$, where $\mathcal{A} = \begin{bmatrix} 0 & T^{\dagger} T & 0 \end{bmatrix}$, with $T$ being a closed and bounded linear relation from a Hilbert space $H$ to a Hilbert space $K$, and $T^{\dagger}$ representing the Moore-Penrose inverse of $T$, the set $\rho(\mathcal{A})$ refers to the resolvent set of $\mathcal{A}$. The paper also explores several interesting results regarding the Moore-Penrose inverses of direct sums of closed relations with closed ranges in Hilbert spaces.